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Properties

Label 1716.2.p.a
Level $1716$
Weight $2$
Character orbit 1716.p
Analytic conductor $13.702$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1716,2,Mod(857,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.857");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.p (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7023289869\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{9} + 48 q^{25} - 12 q^{27} + 48 q^{49} - 16 q^{55} - 12 q^{69} + 44 q^{75} - 40 q^{81} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
857.1 0 −1.65331 − 0.516316i 0 −0.870582 0 3.25162 0 2.46684 + 1.70725i 0
857.2 0 −1.65331 − 0.516316i 0 −0.870582 0 −3.25162 0 2.46684 + 1.70725i 0
857.3 0 −1.65331 − 0.516316i 0 0.870582 0 3.25162 0 2.46684 + 1.70725i 0
857.4 0 −1.65331 − 0.516316i 0 0.870582 0 −3.25162 0 2.46684 + 1.70725i 0
857.5 0 −1.65331 + 0.516316i 0 −0.870582 0 −3.25162 0 2.46684 − 1.70725i 0
857.6 0 −1.65331 + 0.516316i 0 −0.870582 0 3.25162 0 2.46684 − 1.70725i 0
857.7 0 −1.65331 + 0.516316i 0 0.870582 0 −3.25162 0 2.46684 − 1.70725i 0
857.8 0 −1.65331 + 0.516316i 0 0.870582 0 3.25162 0 2.46684 − 1.70725i 0
857.9 0 −1.32610 − 1.11421i 0 −3.23239 0 −2.02885 0 0.517092 + 2.95510i 0
857.10 0 −1.32610 − 1.11421i 0 −3.23239 0 2.02885 0 0.517092 + 2.95510i 0
857.11 0 −1.32610 − 1.11421i 0 3.23239 0 2.02885 0 0.517092 + 2.95510i 0
857.12 0 −1.32610 − 1.11421i 0 3.23239 0 −2.02885 0 0.517092 + 2.95510i 0
857.13 0 −1.32610 + 1.11421i 0 −3.23239 0 2.02885 0 0.517092 − 2.95510i 0
857.14 0 −1.32610 + 1.11421i 0 −3.23239 0 −2.02885 0 0.517092 − 2.95510i 0
857.15 0 −1.32610 + 1.11421i 0 3.23239 0 −2.02885 0 0.517092 − 2.95510i 0
857.16 0 −1.32610 + 1.11421i 0 3.23239 0 2.02885 0 0.517092 − 2.95510i 0
857.17 0 −0.803587 − 1.53436i 0 −2.01342 0 −2.54446 0 −1.70850 + 2.46598i 0
857.18 0 −0.803587 − 1.53436i 0 −2.01342 0 2.54446 0 −1.70850 + 2.46598i 0
857.19 0 −0.803587 − 1.53436i 0 2.01342 0 −2.54446 0 −1.70850 + 2.46598i 0
857.20 0 −0.803587 − 1.53436i 0 2.01342 0 2.54446 0 −1.70850 + 2.46598i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 857.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
39.d odd 2 1 inner
143.d odd 2 1 inner
429.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.2.p.a ✓ 56
3.b odd 2 1 inner 1716.2.p.a ✓ 56
11.b odd 2 1 inner 1716.2.p.a ✓ 56
13.b even 2 1 inner 1716.2.p.a ✓ 56
33.d even 2 1 inner 1716.2.p.a ✓ 56
39.d odd 2 1 inner 1716.2.p.a ✓ 56
143.d odd 2 1 inner 1716.2.p.a ✓ 56
429.e even 2 1 inner 1716.2.p.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.p.a ✓ 56 1.a even 1 1 trivial
1716.2.p.a ✓ 56 3.b odd 2 1 inner
1716.2.p.a ✓ 56 11.b odd 2 1 inner
1716.2.p.a ✓ 56 13.b even 2 1 inner
1716.2.p.a ✓ 56 33.d even 2 1 inner
1716.2.p.a ✓ 56 39.d odd 2 1 inner
1716.2.p.a ✓ 56 143.d odd 2 1 inner
1716.2.p.a ✓ 56 429.e even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1716, [\chi])\).